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Theorem oteq2 4229
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
oteq2  |-  ( A  =  B  ->  <. C ,  A ,  D >.  = 
<. C ,  B ,  D >. )

Proof of Theorem oteq2
StepHypRef Expression
1 opeq2 4220 . . 3  |-  ( A  =  B  ->  <. C ,  A >.  =  <. C ,  B >. )
21opeq1d 4225 . 2  |-  ( A  =  B  ->  <. <. C ,  A >. ,  D >.  = 
<. <. C ,  B >. ,  D >. )
3 df-ot 4042 . 2  |-  <. C ,  A ,  D >.  = 
<. <. C ,  A >. ,  D >.
4 df-ot 4042 . 2  |-  <. C ,  B ,  D >.  = 
<. <. C ,  B >. ,  D >.
52, 3, 43eqtr4g 2533 1  |-  ( A  =  B  ->  <. C ,  A ,  D >.  = 
<. C ,  B ,  D >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379   <.cop 4039   <.cotp 4041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-ot 4042
This theorem is referenced by:  oteq2d  4232  efgi  16610  efgtf  16613  efgtval  16614  el2wlkonot  24692  el2spthonot  24693  frg2wot1  24881  usg2spot2nb  24889  mapdh9aOLDN  36989  hdmap1eulemOLDN  37023
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